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Green Coordinates

What for ?. Green Coordinates. Deforming 2D/3D objects. boundary + inside . Y. Lipman D. Levin D. Cohen-Or. Deform. 2D image. How to Deform?. Cage based space warping A deformed point is represented as a relative position with the cage. Cage 2D : polygon

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Green Coordinates

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  1. What for ? Green Coordinates • Deforming 2D/3D objects. • boundary + inside Y. Lipman D. Levin D. Cohen-Or Deform 2D image

  2. How to Deform? • Cage based space warping • A deformed point is represented as a relative position with the cage. Cage2D : polygon 3D : polyhedron Coordinates 3D mesh

  3. Conventional Coordinates Coordinates : Linear combination of cage vertices

  4. An Example(Barycentric coordinates on simplex) Triangulated cage

  5. Problems of Conventional Methods • Shape-preservation is difficult • Each of (x,y,z) is deformed individually. Proposed Conventional

  6. Previous Work for Deformation • Surface based. + Shape-preservation - Must solve (non-)linear equations. • Space warping based. + Computationally efficient - Difficult to preserve shape Contribution of this paper:Shape-preserving deformation using Space warping.

  7. Coordinates : Proposed Method Coordinates using normals

  8. Interpolation vs. Approximation • It is not possible to achieve to conformal mapping using interpolation methods. Interpolation(not conformal) Approximation(conformal) All coordinates are possibly non-zero. Coordinates : Sometimes negative.

  9. Harmonic Function • As flat as possible • If the boundary condition is linear, then the linearity is reproduced. • Necessary condition of a conformal mapping Solve PDE : Boundary conditions 1D case

  10. Solution using Green’s Identity (In textbook part) (Impulse function) Green’s function of Laplacian : Next page

  11. Solution using Green’s Identity Polygonize the boundary Barycentric coordinates

  12. Calculation of Green Coordinates cage

  13. Properties of Green Coordinates Linear reproduction→ Translation invariance→ Rotation and scale invariance→ We need an appropriate scaling of normals. (Next page) Shape preservation→ 2D : Conformal (Proved in the other paper)→ 3D : Quasi-conformal (Seems to be checked experimentally. Max. distortion = 6) Smoothness→ {φ} and {ψ} are smooth and harmonic

  14. Scaling of Normals Conformal

  15. Extension to Cage’s Exterior Interior: Exterior: → Try to extend the coordinates through a specified face (exit face).

  16. cage How to Extend ? Should keep also in exterior : Two equations in 2D case Make unknown, then decide them by solving a linear system

  17. Multiple Exit Faces • Separate the space. Separation curve Conformingin the cage

  18. Discussions • Only for deformation ? • FEM basis ? • Interpolating a target shape ?(non-linear system must be solved) • Conforming two adjacent cages ? 　　→ Maybe NO.

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